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Difference between revisions of "Demoulin surface"

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A surface formed by a conjugate net of lines the tangents to which form two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309801.png" />-congruences — a so-called singular conjugate system. Only the Demoulin surfaces permit a [[Projective deformation|projective deformation]]. Introduced by A. Demoulin .
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A surface formed by a conjugate net of lines the tangents to which form two $W$-congruences — a so-called singular conjugate system. Only the Demoulin surfaces permit a [[Projective deformation|projective deformation]]. Introduced by A. Demoulin .
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309802.png" /> et les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309803.png" />"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 590–593</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309804.png" /> et les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309805.png" />"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 705–707</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309806.png" />"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 797–799</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309807.png" />"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 927–929</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.P. Finikov,  "Projective-differential geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces $R$ et les surfaces $\Omega$"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 590–593</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces $R$ et les surfaces $\Omega$"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 705–707</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces $R$"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 797–799</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  A. Demoulin,  "Sur les surfaces $\Omega$"  ''C.R. Acad. Sci. Paris'' , '''153'''  (1911)  pp. 927–929</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.P. Finikov,  "Projective-differential geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The terminology concerned with Demoulin surfaces differs. In [[#References|[a1]]] they are roughly characterized by the fact that the Demoulin tetrahedron (see [[Demoulin quadrilateral|Demoulin quadrilateral]]) degenerates to one point. The existence of a projective deformation is a more general condition (see [[#References|[a1]]], Par. 119) The problem of projective deformation is related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309808.png" />-congruences, which are special <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030980/d0309809.png" />-congruences (see [[#References|[a1]]] and [[#References|[2]]], [[#References|[3]]]).
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The terminology concerned with Demoulin surfaces differs. In [[#References|[a1]]] they are roughly characterized by the fact that the Demoulin tetrahedron (see [[Demoulin quadrilateral|Demoulin quadrilateral]]) degenerates to one point. The existence of a projective deformation is a more general condition (see [[#References|[a1]]], Par. 119) The problem of projective deformation is related to $R$-congruences, which are special $W$-congruences (see [[#References|[a1]]] and [[#References|[2]]], [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , '''2''' , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.P. Lane,  "A treatise on projective differential geometry" , Univ. Chicago Press  (1942)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , '''2''' , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.P. Lane,  "A treatise on projective differential geometry" , Univ. Chicago Press  (1942)</TD></TR></table>

Latest revision as of 10:12, 12 April 2014

A surface formed by a conjugate net of lines the tangents to which form two $W$-congruences — a so-called singular conjugate system. Only the Demoulin surfaces permit a projective deformation. Introduced by A. Demoulin .

References

[1a] A. Demoulin, "Sur les surfaces $R$ et les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 590–593
[1b] A. Demoulin, "Sur les surfaces $R$ et les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 705–707
[1c] A. Demoulin, "Sur les surfaces $R$" C.R. Acad. Sci. Paris , 153 (1911) pp. 797–799
[1d] A. Demoulin, "Sur les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 927–929
[2] S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[3] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)


Comments

The terminology concerned with Demoulin surfaces differs. In [a1] they are roughly characterized by the fact that the Demoulin tetrahedron (see Demoulin quadrilateral) degenerates to one point. The existence of a projective deformation is a more general condition (see [a1], Par. 119) The problem of projective deformation is related to $R$-congruences, which are special $W$-congruences (see [a1] and [2], [3]).

References

[a1] G. Bol, "Projective Differentialgeometrie" , 2 , Vandenhoeck & Ruprecht (1954)
[a2] E.P. Lane, "A treatise on projective differential geometry" , Univ. Chicago Press (1942)
How to Cite This Entry:
Demoulin surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Demoulin_surface&oldid=14324
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article